big list - Computer Algebra Errors

This error affects all versions of Mathematica from 6 to 8. The result of a function depends on what letter is chosen for argument when calling it. In the simplest case it can be illustrated as follows:

in:

$A[text{x_}]text{:=}sum _{k=0}^{x-1} x$

$A[k]$

$A[z]$

out:

$1/2 (-1 + k) k$

$z^2$

The correct answer is evidently, the later. This behavior affects not only sums but also integrals, so one have to check so that the letter user for the argument not to coincide with the index variable used for definition. In the case of recursion this becomes very difficult. The following example shows that moving a factor not dependent on the index variable out of the sum sign changes the result:

in:

A1[0,x_]:=1
A2[0,x_]:=1

A1[n_,x_]:=Sum[A1[-1 - j + n, x]*Sum[A1[j, k], {k, 0, -1 + x}], {j, 0, -1 + n}]
A2[n_,x_]:=Sum[Sum[A2[j, k]*A2[-1 - j + n, x], {k, 0, -1 + x}], {j, 0, -1 + n}]

A1[1,x]/.x->2
A1[2,x]/.x->2
A1[3,x]/.x->2

A2[1,x]/.x->2
A2[2,x]/.x->2
A2[3,x]/.x->2

A2[1,2]
A2[2,2]
A2[3,2]

out:

2
5
13

2
5
12

2
5
13

soft question - What are your favorite instructional counterexamples?

Here is a useful example of counter-examples in commutative ring theory;

Let $R=P(mathbb{N})$ be the power set of $mathbb{N}.$ It has a ring structure $(R, +, times)$ where $+$ is the symmetric difference of sets and $times$ is the intersection of sets.

Applications:

Obviously, $R$ is a commutative ring with $1$, ($mathbb{N}$ is the $1$).

1) Let $R$ be a commutative ring with $1$ and a multiplicative closed set of $R$. If $R$ is Noetherian (Artinian) ring then $S^{-1}R$ is Noetherian (Artinian). Does the converse hold?

No, it doesn't.

Using the above example, for any prime ideal $p$ of $R$, $R_p$ (the localization at $p$) is Noetherian (Artinian) while, $R$ is not Noetherian (Artinian).

Outline:

Consider P({1}) $subset$ P({1,2}) $subset...$ and $P(mathbb{N}) supset$ P($mathbb{N} setminus${1}) $supset$ P($mathbb{N} setminus${1,2}) $supset ...$ showing that $R$ is neither Noetherian nor Artinian ring.

It is easy to verify that $R_p$ is isomorphic to $mathbb{Z}/2$, hence it is both Noetherian & Artinian. (Every element of $R_p$ is either $0/1$ or a invertible.)

2) Let $R$ be an integral domain (also commutative with $1$), then for every multiplicative closed set of $R$, $S^{-1}R$ is an integral domain, hence for every $R_p.$ Does the converse hold?

By the above example, it doesn't, since $(P(mathbb{N}),+,times)$ is not an integral domain.

ag.algebraic geometry - Extending maps of curves

(I'm happy to work over an algebraically closed field....)

Let $mathcal{C} rightarrow Spec (R)$ be a (flat) family of proper, prestable curves where $R$ is a DVR. Suppose the generic fiber is smooth and the special fiber, $C_0$, is reduced but may be reducible.

Given a finite map of curves $f_0: D_0 rightarrow C_0$ with $D_0$ also prestable, can this be extended to some map on some family?

That is, is there a flat family of proper curves $mathcal{D} rightarrow Spec(R)$ and an $R$-morphism $f: mathcal{D} rightarrow mathcal{C}$ which reduces to $f_0$ on the special fiber?

Perhaps such an extension is possible only after a ramified cover of $Spec(R)$?

If so, can it be arranged that the generic fiber of $mathcal{D}$ is smooth?

career - When to start reviewing

Reviewing is completely voluntary in all senses. I don't know the percentage of research active mathematicians who have ever reviewed for MathReviews or Zentralblatt, but it is certainly bounded away from 100%. It is relatively rare for a research mathematician to write reviews in any quantity over a period of more than a few years: I think it's something that many of us try out for a while but don't stick with in the long run. For a personal touch, I am a 2003 PhD, I have written about five math reviews, and I have gotten overdue notices on about four more. I would rather write nice, insightful reviews for half of the papers that I get sent, but I haven't yet figured out what do about that.

I agree with Tim Porter that, as a graduate student, you should discuss this with your advisor. I think that if any of my students asked me about it, I would mildly discourage them from writing reviews, for the following reasons:

(i) It is not closely enough related to your thesis work to be a good use of a mid-to-late career grad student's time.

(ii) Prospective employers are generally not going to be more excited about your application because of the reviews you have written.

(iii) Most graduate students -- even ones who have already done some research work of their own -- don't have a broad enough perspective to write insightful reviews. (Or, more positively put, they will have a broader perspective later on and will probably write only a bounded number of reviews in their life no matter when they start.)

Finally, I should say that refereeing papers is a totally different story. I feel that it is an ethical (and karmic) responsibility to referee at least as many papers as one submits. A graduate student can make a good referee for some papers, because they are less likely to feel that it is beneath them to read the paper line by line and really make sure it is correct.

noncommutative algebra - Uniqueness of maximal subfields

I think this question presupposes something that is incorrect. Here is an example. Fix a field $F$ of characteristic 2 and put $Z$ for the field $F(a,b)$ where $a$ and $b$ are indeterminates. Define $D$ to be the quaternion division algebra with center $Z$ generated by elements $i, j$ satisfying

$$i^2 = a,quad j^2 + j = b,quad ij = (j+1)i.$$
Let us denote it by $lt a,b]$, as PK Draxl does in his book "Skew Fields". This is a division algebra because $a$ is not a norm from the separable quadratic extension $E$ obtained from $Z$ by adjoining a root of $x^2 + x + b$.

Obviously $D$ contains the purely inseparable quadratic extensions $Z(sqrt{a})$. I claim it also contains the extension $Z(sqrt{ab})$. To see this, we calculate in the Brauer group:

$$lt a,b] = lt ab^2,b] = lt ab,b] + lt b,b] = lt ab, b]$$
where the last equality is because $lt b,b]$ is split, i.e. isomorphic to 2-by-2 matrices, which follows from the fact that $b$ itself is a norm from $E$ (in fact, the norm of the element $x$).

Because $b$ is a nonsquare in $K$, we have found two non-isomorphic purely inseparable quadratic extensions in $D$ of exponent 1.

ag.algebraic geometry - Extending group actions on varieties

Well, not always, e.g. take $X=mathbb P^2$ with homogeneous co-ordinates $x,y,z$, $U$ the locus defined by $xyzne 0$ and $sigma$ the involution of $U$ defined by $(x,y,z)mapsto (yz,xz,xy)$ (the standard quadratic Cremona transformation of $X$). Then $sigma$ is of order $2$ and does not extend to $X$. On the other hand some sufficient conditions, in characteristic zero, are: there are no rational curves in $Xsetminus U$; $X$ has ample canonical class and is smooth (this can be weakened to having canonical singularities).

[Update: Charles Siegel asked for references, and I have none to hand, although this is all well known.]

For the Cremona example, recall that every automorphism of $mathbb P^2$ is linear, and $sigma$ clearly isn't. More geometrically, $sigma$ blows up the vertices of the triangle $xyz =0$ and collapses its sides.

No rational curves in $Xsetminus U$: fix $gin G$ and think of it as a rational map $g:Xto X$. By Hironaka, there is a minimal composite $p:Zto X$ of blow-ups such that $gcirc p$ is regular. If $Xsetminus U$ has no rational curves, then the rational curves in the exceptional divisor of the last blow-up are contracted, so the last blow-up was redundant, contradicting minimality.

Ample canonical class $omega_X$: then $X=Proj(R(X,omega_X))$, with $R(X)= oplus_{nge 0}H^0(X,omega_X^{otimes n})$, the canonical ring. This is a birational invariant of $X$, so $G$ acting on $U$ acts on $R(X)$, so on $X$.

ag.algebraic geometry - Examples of Poisson Schemes

A Poisson Manifold is a real manifold $M$ along with a Lie bracket $[cdot,cdot]$ on $C^infty(M)$ which is a derivation in each variable. Poisson manifolds are interesting for a few reasons, among them:

1. You can define the notion of an integrable system structure on a Poisson manifold, which allows them to be applied to solving problems in physics with enough symmetry.


2. Every symplectic manifold is automatically Poisson


3. Any Poisson manifold has a foliation by symplectic leaves.

(Reference for all of this: anything on Poisson manifolds, in particular, wikipedia.)

Now, I've seen people seriously (for instance, in Vanhaecke's book) extend this notion to affine varieties over $mathbb{C}$, where being Poisson means that the structure sheaf is a sheaf of Poisson Algebras, specifically, a Poisson algebra is an associative algebra along with a Lie bracket that is a derivation in each variable.

Now, what I'm interested in is how far this can be generalized and still have something where there are interesting (new!) examples. For instance, is "Poisson Scheme $X$ over $S$" a real object of interest? Specifically, I'm wondering if there are any examples where $S$ is not the spectrum of a field of characteristic zero, say $S$ is a finite field, or something positive dimensional, or nonreduced, etc, and if there are examples of this form, what makes them interesting? For instance, one reason that Poisson manifolds are interesting is that they are applicable to physics and, in fact in many cases to problems related to the geometry of the moduli space of vector bundles on a Riemann surface.

fa.functional analysis - Topological "Interpolation" ?

I think Ady's question has a negative answer.

Recall that the Mazur map, $T$, from the unit ball of $ell_2$ onto the unit ball of $ell_1$, is defined by
$T(sum a_i e_i)= sum a_i^2 b_i e_i$, where $b_i$ is the sign of $a_i$. It is a uniform homeomorphism in the norm topologies and obviously is coordinatewise continuous, which means that it is weak ($=$ weak$*$ in $ell_2$) to weak$^*$ continuous (since on the unit ball of $ell_p$ with $p$ finite the
weak* topology is the topology of coordinatewise convergence).
$T$ is not weak to weak continuous since the unit vector basis converges weakly to zero in $ell_2$ but not in $ell_1$.

The problem is to extend $T$ to all of $ell_2$. I think that the easiest way to do this is to show that there is a retraction $R$ from $ell_2$ onto its unit ball which is both norm to norm continuous and weak to weak continuous. Define $R$ on the complement of the unit ball by

$$R(sum a_i e_i) = sum_{i=1}^n a_i e_i + t e_{n+1},$$
where $sum_{i=1}^n a_i^2 le 1 < sum_{i=1}^{n+1} a_i^2$ and $t$ is chosen to have the same sign as $a_{n+1}$ and to make the image vector have norm one. $R$ is obviously continuous (even Lipschitz) in the norm topology and is continuous in the topology of coordinatewise convergence, hence is weak to weak continuous (since in all of $ell_2$, the weak topology is stronger than the topology of coordinatewise convergence).

To get a counterexample that maps one dual space to itself, work with the space $ell_2 oplus ell_1$.

nt.number theory - Nonnegative polynomial in two variables

The following theorem of Artin -- his solution of Hilbert's 17th problem, but in a stronger form than Hilbert himself asked for -- answers the question.

Theorem (Artin, 1927): Let $F$ be a subfield of $mathbb{R}$ that has a unique ordering, and let $f(t) = f(t_1,ldots,t_n) in F(t_1,ldots,t_n)$ be a rational function such that
$f(a) geq 0$ for all $a = (a_1,ldots,a_n) in F^n$ for which $f$ is defined. Then $f$ is a sum of squares of rational functions with coefficients in $F$.

A proof can be found in Jacobson, Basic Algebra II, Section 11.4.

Note that the tempting strengthening -- that if $f$ is a polynomial, it is a sum of squares of polynomials -- is false, as Hilbert himself showed.

mp.mathematical physics - Is there a theory of differential equations for smooth correspondences?

This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth manifold and $mathbb R$, say). But I'm wondering about a specific well-studied case.

Background

Let $N$ be a compact smooth manifold with tangent bundle $TN$ and cotangent bundle $T^*N$. In the usual way, pick a Lagrangian $L: TN to mathbb R$, and suppose that it is nondegenerate in the sense that $frac{partial L}{partial v}$, thought of as a map $TN to T^*N$, is a fiber bundle isomorphism, where $v$ is a (vector of) fiber coordinate(s). Then, in the usual way, we can define a Hamiltonian $H: T^*N to mathbb R$ by $H = pv - L$, where $pv$ is the canonical pairing between a cotangent vector and a tangent vector, and I use $frac{partial L}{partial v}$ to identify $TN$ with $T^*N$. In the usual way, define on $N$ the second-order ordinary differential equation $frac{d}{dt}bigl[ frac{partial L}{partial v}bigr] = frac{partial L}{partial q}$ and $v = frac{dq}{dt}$, or equivalently $frac{dq}{dt} = frac{partial H}{partial p}$ and $frac{dp}{dt} = -frac{partial H}{partial q}$. (These are all written in local coordinates, where $q$ is a local coordinate on $N$ and $v = dq$ and $p = frac{partial}{partial q}$ are the corresponding coordinates on $TN$ and $T^*N$. But the ODE is coordinate-invariant.)

Since $N$ is compact and $L$ is nondegenerate, the ODE has global solutions, and each solution is determined by its initial conditions, which are given equivalently by a point in $TN$ and a point in $T^*N$. In the usual way, define an action map $S: TN times mathbb R to mathbb R$ by $S(v,q,t) = int_0^t L(phi(v,q,s))ds$, where $phi: TN times mathbb R to TN$ is the "flow" map for the ODE.

Let $pi$ be the projection $TN to N$, so that $pi(v,q) = q$, and using the flow map $phi$, define a map $TN times mathbb R$ to $N times N times mathbb R$ via $(v,q,t) mapsto (q,pi(phi(v,q,t)),t)$. Generically, this map is a local isomorphism, in the following sense: for generic $(v,q,t)$ (say, a dense open subset of $TNtimes mathbb R$ when $frac{partial^2 L}{partial v^2}$ is positive-definite), there is a small open neighborhood such that the map to $Ntimes N times mathbb R$ takes the neighborhood diffeomorphically to its image. Pick one such small neighborhood, and use it to push forward the action function $S$. Abusing notation, I will call this pushforward $S$. Then $S(q_1,q_2,t)$ satisfies the Hamilton-Jacobi equation: $frac{partial S}{partial t} = - H(frac{partial S}{partial q_2},q_2) = - H(-frac{partial S}{partial q_1},q_1)$.

My question

Above, I defined a local function $S: U to mathbb R$, where $U$ is a small neighborhood of $Ntimes Ntimes mathbb R$, and it satisfied a partial differential equation. But really I should have talked about the correspondence

$$Ntimes Ntimes mathbb R overset{(pi, picirc phi, t)}{longleftarrow} TN times mathbb R overset{S}{longrightarrow} mathbb R$$

Is there language with which one can say that this correspondence satisfies the Hamilton-Jacobi equations? For example, what happens near non-generic (sometimes called "focal") points?

Bonus questions

(If $N$ is not compact, then the flow map does not have global-time solutions. But I can still do everything; I just have to replace the space $TN times mathbb R$ by an open subspace. In fact, $phi$ still defines on $TN times mathbb R$ the structure of an action groupoid, and both $(pi, picirc phi, t)$ and $S$ are groupoid homomorphisms, so that the above correspondence is a span of groupoids. Does this enter the discussion in any interesting way?)

If $frac{partial L}{partial v}$ does not define a bundle isomorphism, then the Hamiltonian is not well-defined as a function $H: T^*Nto mathbb R$. But it does make sense as a correspondence, by:

$$T^*N overset{ frac{partial L}{partial v}}{longleftarrow} TN overset{ vfrac{partial L}{partial v} - L}{longrightarrow} mathbb R$$

Imposing the condition that $frac{partial^2 L}{partial v^2}(v,q)$ is an invertible matrix for each $(v,q)$, so that the ODE is still nondegenerate second-order, does the language of correspondences allow me to talk about the Hamilton-Jacobi equations when the Hamiltonian is not a function but the above correspondence?

Alternately, I could start with a function $H: T^*N to mathbb R$, and construct a correspondence $L$, etc. The Legendre transform should really be thought of as a transformation of correspondences, not of functions.

Finally, when $frac{partial^2 L}{partial v^2}$ is sometimes degenerate, then the ODE degenerates, sometimes in complicated ways. Can this be accommodated by making the flow $phi$ into a correspondence rather than a function?

modular rep theory - Exact sequences of permutational representations?

Now that I have thought about the question some more, I can give a better answer. I have a remark about how to search for these resolutions in general, and a construction that leads to many examples.

First, every permutation module is part of many exact sequences of permutation modules in which the subgroup is trivial:

$$cdots to R[G]^{n_2} to R[G]^{n_1} to R[G/H] to 0.$$
The reason is very simple and standard: Any exact complex of this type is by definition a free resolution. The way that you make a free resolution is that there is some intermediate target or kernel $K$, and you can send the generators $1 in R[G]$ to some spanning set of $K$. Usually the resolution is infinite, but with standard linear algebra you can search for a finite solution when there is one.

This observation generalizes to other permutation modules. There is a part of induction-restriction reciprocity that holds over any ring. Namely,

$$text{Hom}_G(R[G/H],M) cong text{Hom}_H(I,M),$$
where $I$ is the trivial representation. This relation is a generalization of the proof that a free module is projective. So if there is a finite permutation resolution of a module $M$ (which could be the kernel of some incomplete sequence of permutation modules), you can search for it in the same way that you search for free resolutions.

Second (and I suspect that readers will like this answer better), you can obtain many examples from the chain homology complex of a finite CW complex $K$ with an action of $G$. In order to make everything match, let's consider a slight generalization of a permutation module, not just $R[G/H]$ but also a module $R[G/H]_chi$ induced from a character

$$chi:H to {1,-1}.$$
The basic idea is not hard: Each term $C_n(K)$ is a direct sum of signed permutation modules of $G$, because $G$ acts on the cells. If a cell $c$ has stabilizer $H$, then it makes an orbit equivalent to $G/H$, and we can define $chi$ by examining which elements of $H$ flip over $c$. If $K$ happens to have the same $R$-homology as a point, then you can augment its chain complex by the trivial module. Or if it is an $R$-homology sphere, you can augment its chain complex at both ends. If you don't like the signed permutation modules, you can subdivide the cell $c$ to get rid of them, or work in characteristic 2.

If $K$ is a line segment and $G = C_2$ acts by reflecting it, the result is Leonid's first example.

If $K$ is a polygon with $n$ sides and $G = C_n$ acts by rotation, the result is Leonid's second example.

If $K$ is a polygonal tiling of the 2-sphere and $G$ is a rotation group that acts on $K$ without reversing edges, the result is a new example. For instance you can take a dodecahedron graph and divide each edge into two edges. Again, the point of splitting the edges is just to get rid of the signed permutation modules.

Every finite group $G$ acts faithfully on a sphere of some dimension, because $G$ has a faithful linear representation. So there are many sphere examples for every finite group.

At first glance, the second answer is a type of construction. In many cases, it is also an interpretation of a chain complex $C$, because if you have $C$ you can try to build a CW complex to represent it. In order to be an augmented chain complex, $C$ needs to end in the trivial module $I$. If it ends in something else, you can concatenate with

$$0 to I to I to 0.$$
The differentials of $C$ also need to have integer matrices.

---

I found several papers on a related question called the "quasi-projective dimension" of a group ring $R[G]$. The original paper on this is Groups of finite quasi-projective dimension, by Howie and Schneebeli. Their definition of a quasi-projective resolution is a finite resolution of a module $M$ by projective terms, and at the end a term which is a permutation module. I assume that, certainly for finite groups, it would work just as well to use a free resolution as a projective resolution. Among other results, Howie and Schneebeli establish that if $G$ is a finite group and $R = mathbb{Z}$, then the quasi-projective dimension of $R[G]$ equals the period of its Tate cohomology. But another theme of the paper is that these questions, both theirs and surely Leonid's also, are perfectly interesting for infinite groups too.

The papers that cite this initial paper use the second idea that I propose above. They make CW complexes with an action of the group $G$, and then make chain complexes from these CW complexes. So these CW complexes seem like a main way to understand complexes of permutation modules. In my opinion, the CW complex picture suggests generalizing the original question to include signed permutation modules.

geometry - Finding Constant Curvature Metrics on Surfaces without full power of Uniformization

(I rewrote this question, hopefully it's more clear now. It's still the same question, but I reordered its parts.)

Let S be a surface (possibly non-compact, but no boundary). It seems that there are three different theorems that people sometimes call "the Uniformization Theorem":

1. There is some constant-curvature metric on S.




2. Given any conformal structure on S, there is a constant-curvature complete metric on S that represents it. More precisely,

   
   
   

   
   

   Given any metric g on S, there is a (unique) complete metric g₀ of constant curvature that represents the same conformal structure, i.e. there is a strictly positive function $fin C^infty (S)$ such that g'=fg.

   
   




3. Up to diffeomorphism, there is only one conformal structure on the sphere, and two on a topological disk (namely, the disk with the standard hyperbolic metric or the plane with the Euclidean metric).

The first of these is easy, see proof below. It's also easy to see that 3 implies 2, see below.

Henceforth, "Uniformization Theorem" will mean number 3. It is very powerful, but also rather mysterious. Its proofs are rather complicated and usually involve solving a somewhat complicated differential equation, here is a very interesting discussion about that.

I feel that 2, on the other hand, should be simpler, and not require the full power of uniformization to prove it. I also think it is quite distinct in spirit from 3, and so having a different proof for it could be illuminating. So, my basic questions are:

• Are there any nice proofs of 2. that do not use 3.?


• Is there any way to prove 3. from 2.? (this would imply that the answer to the above question is no. This is certainly possible unless we remove the work "unique" from the statement of 2., so we'll do that)

(It would be nice if the proof worked for non-compact surfaces, i.e. with cusps or funnels. Then, however, it could be tricky to make the metric complete - or maybe there is some trick to make any constant-curvature metric complete?)

---

What follows are the promised proofs together with some vague thoughts on how to prove number 2 using them or otherwise. Ideally, it would be great to have two proofs, one for Ricci flow, and another one with something like the proof of theorem 1. Also, perhaps there are other/better approaches?

Proof of theorem 1. above (used to be Proof #2, sorry)

Here's a proof that just constructs some constant-curvature metric on any compact surface. This basically comes from Thurston's book, and could probably be adapted to surfaces with boundary, although it wouldn't be trivial.

Namely, a genus-g surface can be thought of as a regular 4g-gon with sides glued in the appropriate fashion. (If g=0, it's already a sphere, so we ignore this case). For example, a torus is a square with opposite sides glued. So, we can tile the plane with squares, by taking one square and translating it around so that new squares are adjacent to old ones.
We notice that these translations do not change the standard metric, and conclude that the plane modulo a $mathbb Z^2$ action by translations is a torus with a constant-curvature metric.

Similarly, if $g>1$, we can present a regular 4g-gon in the hyperbolic plane. By varying its size, we can make the angles at vertices as small as we want (if it's tiny, the angles will be almost as large as they would be for a Euclidean regular 4g-gon. If it's so huge that the vertices are almost at the line at infinity, the angles will be almost zero. So, we can get any intermediate value). If we make these angles precisely $frac{2pi}{4g}$, then translating the 4g-gons around just like we translated squares before will give us a tiling of the hyperbolic plane. Just like before, quotienting out the hyperbolic plane by the appropriate group will give us our surface with a constant-curvature metric.

Note that we can get many other conformal structures on our surface by replacing our regular *4g*-gon with an irregular one (we can pick any length for each side, I think, as long as the sides we glue have the same length). However, it's not clear that we could get any conformal structure this way. So, this is also a vague approach to proving number 2.

Proof idea for number 2: Ricci-like flow (used to be Proof idea #1)

It seems that we could try to apply something like the Ricci flow to the metric g on S to make its curvature uniform. I don't know enough analysis to fill in the details; I'm wondering whether it's possible to create such a flow so that it does not change the conformal structure. Also, getting a complete metric could be tricky.

(Many people answered that this is possible. In particular, Dmitri's answer pretty much settled for me this part of the question. Only one small question remains: will these flows work for non-compact surfaces and result in a complete metric? Also, it's not quite clear to me whether the proofs with flows are powerful enough to show #3. My guess would be that they aren't, but Hamilton's paper seems to claim that the Ricci flow proof is.)

Appendix: Proof of 2. from Uniformization (#3.) (Used to be Proof #0)

Consider the universal cover $U=tilde S$ of S with the metric g pulled back from S. By uniformization, there is a conformal map $f: (U,g) to (X,g_{standard})$ where $(X,g_{standard})$ is one of three: the Poincare disk with the standard hyperbolic metric, the plane with the Euclidean metric, or the sphere with the standard spherical metric. In any case, it's a standard space with a constant-curvature metric.

As Tom's comment pointed out, at least in the hyperbolic case, all conformal maps on the disk preserve the constant-curvature metric (we can list what they all are). Since the covering transformations become conformal maps on X, they preserve the metric. So, we can pull $g_{standard}$ back to $U$, and then down to $S$ to get $g_0$. The non-hyperbolic cases can also be dealt with.

Usual disclaimer: there might be mistakes, point them out. Thank you!

nt.number theory - How good is Kamenetsky's formula for the number of digits in n-factorial?

A counterexample is $n_1 := 6561101970383$, with

$$log_{10} left( (n_1/e)^{n_1} sqrt{2pi n_1} right) = 81244041273652.999999999999995102483 - phantom; ,$$
but
$$log_{10} (n_1!) = 81244041273653.000000000000000618508 + phantom;.$$
If I computed correctly, $n_1$ is the first counterexample, and the only one up to $10^{13}$. The computation should reach $10^{15}$ sometime next week, with a probability of about $1 - exp(-frac16) sim 15%$ of finding an $n_2$.

The computation (in gp/pari) took about 40 CPU hours here, compressed to 4 hours by running in parallel on 10 of the 12 heads of alhambra.math.harvard.edu . This was not done by calculating $log_{10} (n!)$ to enough precision for every $n leq 10^{13}$, which would have taken hundreds of times longer. The problem of finding nearly integral values of $log_{10} (n!)$ is a special case of the "table maker's dilemma" (Wikipedia attributes this felicitous coinage to William Kahan); in this case, the linear-approximation technique suggested by Lefèvre at the bottom of page 15 of his slides takes time $tilde O(N^{2/3})$ to find all examples with $n < N$. That's what's running on alhambra now.

Along the way a few more terms of sequence A177901 turned up:
$252544447$,
$1430841730$,
$5042264463$,
$31774693500$,
$40752166709$,
$46787073630$,
$129532358256$,
$421559495894$,
$2418277169072$,
$6105111564681$,
and then $n_1 = 6561101970383$, which might even turn out to be the last term up to $10^{15}$ because $log_{10} (n_1!)$ is so close to an integer (about $9$ times closer than necessary for our purpose). [EDIT It's the last term $<10^{14}$ but not $10^{15}$, see below.]
The term $252544447$ was reported on math.se #8323 by Byron Schmuland [EDIT and a few months earlier by David Cantrell on sci.math], though it has not been posted to OEIS yet. The further ones seem to be new, and I'll post them on OEIS soon.

Kamenetsky was right to suggest that the approximation should fail sometimes: in base 10, we expect $n$ to be a counterexample with probability about $1/cn$ with $c = 12 log 10$, so on average each range $[N, 10^{12}N]$ should have about one. Thus it is not surprising that the first one (past $n=1!$) turns out to have $13$ digits. This heuristic is also the source of the estimate $1-exp(-frac16)$ for the probability of another counterexample in $[10^{13}, 10^{15}]$.

UPDATE The calculation has now passed $10^{14}$, finding no new counterexample. It did, however, find a new term for the OEIS sequence a bit beyond $10^{14}$: $n=125291661119688$, with $log_{10}(n!)$ close to but just below the nearest integer $1711938609606982$ (where a counterexample must be a bit above), and also not quite as close as $1/(12n)$ — the difference is about $1/(8.4n)$.

While I'm at it: I should have mentioned that the gp/pari computation also found (in a minute or two) all the terms in $[10^4,10^8]$ listed by OEIS, which lends the new results some credibility; and I thank Gerry Myerson for drawing my attention to this question with his edit of about two weeks ago.

soft question - Theory mainly concerned with $lambda$-calculus?

Automata theory is mainly concerned with Turing machines and all its relatives-in-spirit. $lambda$-calculus is rather rarely mentioned in textbooks on automata theory.

What's the common name of the theory mainly concerned with $lambda$-calculus and its relatives? (I think, "mathematical logic", "computability theory", "programming language theory" and "recursion theory" are too general, compared to "automata theory". But there should be an "$lambda$-theory", shouldn't it?)

sg.symplectic geometry - To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?

Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.

0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,omega)$, let be given a lagrangian foliation $mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $omega$.
(Instead, I mean a lagrangian fibration of $(M,omega)$ as a surjective summersion $f:Mto B$ whose fibers are lagrangian w.r.t. $omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)

1.Local Existence of lagrangian submanifolds transversal to $mathcal{F}$.
For any $pin M$, there exists a lagrangian submanifold of $(M,omega)$ which passes through $p$ and is transversal to $mathcal{F}$.

Infact, for any $pin M$, there exists a chart $(U,phi)$ for $M$ centered at $p$, such that:
$omega= sum_{i=1}^{n}{dphi_i wedge dphi_{n+i}}$,
the restriction of $mathcal{F}$ on $U$ is generated by $frac{partial}{partialphi_{n+1}},ldots,frac{partial}{partialphi_{2n}}$,
and consequently $phi_{n+1}=ldots=phi_{2n}=0$ is a local lagrangian submanifold of $(M,omega)$ passing through $p$ and transverval to $mathcal{F}$.

This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $dphi_1,ldots,dphi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $mathcal{F}$.

2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,omega)$, is transversal to $mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $tau_L^{ast}:T^{ast}Lto L$,
$f_{ast}omega$ is the canonical symplectic on $T^{ast}L$,
and $f$ takes the leaves of $mathcal{F}$ in the fibers of $tau^{ast}_L$.

This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.

ag.algebraic geometry - Pole of a function in a diagonal embedding of a normal affine variety.

Let $X$ be a normal affine algebraic variety of dimension $n$ and ${bar X}^1$, ${bar X}^2$ be complete normal varieties containing $X$ as a (Zariski) dense open subset. Let $f$ be a regular function on $X$ such that $f$ has a pole along every irreducible component (which is necessarily of dimension $n-1$) of $bar X^i setminus X$ for each $i$, $1 leq i leq 2$. Let $bar X subseteq {bar X}^1 times {bar X}^2$ be the closure of the diagonal embedding of $X$ into ${bar X}^1 times {bar X}^2$. Is it true (assuming that $bar X$ is also normal) that $f$ has a pole along every irreducible component of $bar Xsetminus X$?

It is easy to see that this is true when $n=2$ or when $X$ is a (Zariski) open subset of a torus $mathbb T$ and ${bar X}^i$'s are toric completions of $mathbb T$. But the general case still eludes me.

big list - Nontrivial question about Fibonacci numbers?

Here you have some of the coolest ones I have heard of:

1) Let $a$ be a positive integer. Then $a$ is a Fibonacci number if and only if at least one member of the set {$5a^{2}-4, 5a^{2}+4$} is a perfect square.

I think the result is original with Prof. Ira Gessel.

2) Let $phi$ denote the Euler totient function. Prove that $phi(F_{n}) equiv 0 pmod{4}$ if $n geq 5$.

The proof consists of an unexpected application of Lagrange's theorem in Group Theory. Guess there are some other ways to prove it, but that approach will always remain my cup of tea. The problem was posed and solved in the Monthly in the 70's (if my memory serves me right). Look for all entries by Clark Kimberling in that magazine and you'll surely find it.

3) Can you find $(a,b,c) in mathbb{N}^{3}$ such that $2 < a < b < c$ and $F_{a} cdot F_{b} = F_{c}$?

This problem would be trivial if instead of the $cdot$ we had placed a plus sign there. In any case, there is no need to panic with this proposal. All you need to recall is the corresponding primitive divisor theorem.

4) Ben Linowitz mentioned above a beautiful result by Professor Florian Luca, namely:

There aren't any perfect numbers in the Fibonacci sequence.

I read the paper in my junior year and I didn't find it that hard to follow. The easy part of this cute note resides in the proof of the fact that there are no even perfect numbers in the Fibonacci sequence. Guess this result is interesting enough to deserve consideration in those lectures that you intend to give. If this proposal is not exactly your idea of excitement, you can take a look at some of the other papers by Professor Florian. He writes a lot about recurrence sequences. Another theorem of his, closely related with the subject matter of this discussion, ascertains that

There is no non-abelian finite simple group whose order is a Fibonacci number.

5) Last but not least... Prove that the sequence {$F_{n+1}/F_{n}$}$_{n in mathbb{N}}$ converges and use this fact to derive the continued fraction development for the golden ratio.

This one should be well-known, yet it would be nice to see what your students come up with...

Added (Nov 20/2010) I've just noticed that the Fibonacci Assn. has made available the articles published in The Fibonacci Quarterly between 1963 and 2003. I'm sure you will find plenty of additional material among those files that they have so generously released for our enjoyment. For instance, the seminal paper by J. H. E. Cohn that K. Buzzard mentions below can be found here.

cv.complex variables - Irrational Numbers and the Riemann Surface of a Multi-Valued Function

EDIT 1/5/2010: I was a little dissatisfied with the informality of what I wrote below, so here is a somewhat more formal writeup. The statement is somehow geometrically obvious, but the proof is still a bit nice.

EDIT: I initially claimed the sequence below is short exact, which is false; it is right exact. It is fixed below, with some explanation, and a bit of a geometric explanation of what's going on.

I know little about the theory of linear independence over $mathbb{Q}$, but I'll attempt an answer to this part of the question:

Are there relationships between the (co)homology groups of the covering and the residues?

The answer is "yes." Let $f$ be a meromorphic function on $mathbb{C}$, and for convenience let's assume that it has poles $z_1, ..., z_n$ of order $1$, and no other singularities. Let $g$ be an antiderivative of $f$. Then the Riemann surface of $f$ is $M_f=mathbb{C}-{z_1, ..., z_n}$ and the Riemann surface of $g$, which we will denote by $M_g$, is a covering space of $M_f$ with covering map $pi: M_gto M_f$. Let $Vsubset mathbb{C}$ be the $mathbb{Q}$-vector space spanned by $operatorname{Res}_{z_1}(f), ..., operatorname{Res}_{z_n}(f)$. Then there is a short exact sequence

$$H_1(M_g, mathbb{Q})overset{H_1(pi)}{longrightarrow} H_1(M_f, mathbb{Q})overset{int}{longrightarrow} Vto 0,$$
where the map $int$ is given as follows. Namely, $int: H_1(M_f, mathbb{Q})to V$ is given by $[gamma]mapsto frac{1}{2pi i}int_gamma f~operatorname{dz}$.

Let's elucidate the connection to the linear independence of $operatorname{Res}_{z_1}(f), ..., operatorname{Res}_{z_n}(f)$ over $mathbb{Q}$. $H_1(M_f, mathbb{Q})$ is a $mathbb{Q}$-vector space with a basis of cycles $[lambda_1], ..., [lambda_n]$ corresponding to the punctures $z_1, ..., z_n$. Then the map $int$ sends $[lambda_i]$ to $operatorname{Res}_{z_i}(f)$. So the image of $H_1(M_g, mathbb{Q})$ in $H_1(M_f, mathbb{Q})$ is precisely the vector space of relations between the residues of $f$.

Added: We can extend this right exact sequence into a longer sequence. In particular, by covering space theory we have that $pi_1(M_g)to pi_1(M_f)$ is an injection. It is easy to see that the commutator subgroup of $pi_1(M_f)$ is contained in the image of $pi_1(M_g)$. By the Hurewicz theorem

$$H_1(M_g, mathbb{Q})simeq pi_1(M_g)^{Ab}underset{mathbb{Z}}{otimes} mathbb{Q}.$$
So the kernel of the map $H_1(M_g, mathbb{Q})to H_1(M_f, mathbb{Q})$ is given by the image of $[pi_1(M_f), pi_1(M_f)]$ (which is contained in $pi_1(M_g)$) in $H_1(M_g, mathbb{Q})$. One can extend the exact sequence further back by looking at quotients of commutators in this manner.

This first extension has a geometrical interpretation. Namely, let $h$ be a meromorphic function whose poles have the same locations as those of $f$, but whose residues are linearly independent over $mathbb{Q}$. Then the antiderivative of $h$, denoted $s$ has Riemann surface $M_s$, which is a covering space over $M_f$, with covering map $pi': M_sto M_f$. By the properties of covering spaces, $pi'$ factors through $pi$, and it is not hard to see that $pi_1(M_s)$ is exactly the commutator subgroup of $pi_1(M_f)$. Then the sequence

$$H_1(M_s, mathbb{Q})overset{H_1(pi')}{longrightarrow} H_1(M_g, mathbb{Q})overset{H_1(pi)}{longrightarrow} H_1(M_f, mathbb{Q})overset{int}{longrightarrow} Vto 0,$$
is exact, and coincides with the sequence described above.

I don't know if the continuing left extensions of this sequence have similar geometric interpretations. Also, it would be nice to have a naturally arising description of this sequence, rather than the somewhat ad hoc one I've given.

nt.number theory - Dirichlet and the prime number theorem

Dirichlet's remark from the first paper is extracted and translated on page 98 of The Development of Prime Number Theory by Narkiewicz. So this has not passed completely unnoticed. Narkiewicz remarks that Dirichlet believed that his analytic methods would enable him to prove Legendre's conjecture, and that Dirichlet never returned to the problem.

Dirichlet remained interested in the asymptotic growth laws ("Asymptotische Gesetze") of arithmetic functions for the rest of his life, as seen from his 1849 paper with the estimate

$$sum_{n leq x}d(n) = xlog(x) + (2gamma - 1)x + O(x^{1/2}),$$

and a couple of other estimates, and a letter of 1858 to Kronecker reprinted in Dirichlet's Werke, where he mentions having obtained a substantial improvement of the error term $O(x^{1/2})$ by a new method.

Since Dirichlet demonstrably did not lose interest in such questions, and never returned to the PNT in print, it seems reasonable to believe that he discovered that his real-variable method would not yield the PNT.

ag.algebraic geometry - Proof of Borel-Weil-Bott Theorem

The simplest proof of Borel-Weil-Bott that I know is due to Demazure: he has two papers in Inventiones (one in 1968 the other in 1976) on the theorem, and the second is two pages long -- it gives a simplification of his previous proof, and he uses only algebro-geometric techniques. Both papers are readable.

Right actions of operads and monads

I hadn't seen this question before. Better late than never, perhaps. I'll give some categorical background first, which may well be familiar, but to get everything sorted out we should probably recall it anyway.

A more than sufficient set of hypotheses is that we're working in a cocomplete closed symmetric monoidal category $V$. Let $mathbb{P}$ be the permutation category. There is a well-known monoidal product $boxtimes$ on $V^{mathbb{P}^{op}}$ called "substitution product", given objectwise by the formula

$$(F boxtimes G)(n) = sum_k F(k) otimes_{S_k} G^{otimes_{Day} k}$$

where $otimes_{Day}$ is the Day convolution product. (The more standard notation is $circ$ instead of $boxtimes$, but that is potentially confusing here.) Monoids in the monoidal category $(V^{mathbb{P}^{op}}, boxtimes)$ are the same thing as operads. The nLab article on operads provides plenty of explanation and background for this view on operads.

And, as in any monoidal category, a monoid $A$ induces a monad structure on the monoidal category, in fact two monad structures: a left one $A boxtimes -$ whose algebras are left $A$-modules, and a right one $- boxtimes A$ whose algebras are right $A$-modules. This applies in particular to operads $A$, so we have two monads, $L_A$ and $R_A$ respectively, both acting on $V^{mathbb{P}^{op}}$.

Then a right action of an operad on a functor $F: mathbb{P}^{op} to V$ is the same thing as an (ordinary, left-sided) algebra/module of $R_A = - boxtimes A$. But you can also consider a right action as inducing a right module $F boxtimes -$ over the monad $L_A = A boxtimes -$. So the answer to your question is certainly 'yes', but you'll have to decide for yourself how interesting the answer is.

Perhaps I could say a few more things. In your notion of right action on an object $X$ of $V$, what you essentially did is embed $V$ in $V^{mathbb{P}^{op}}$ by sending an object $X$ to the evident $mathbb{P}$-representation $X^{otimes bullet}$, so that your notion of right action on $X$ takes the form of a right $A$-module

$$(X^{otimes bullet}) boxtimes A to X^{otimes bullet}$$

in the category $V^{mathbb{P}^{op}}$. That's fine, but I'll note that that choice of embedding doesn't quite match the one used for the usual notion of (left) algebra over an operad. For this, we embed $V$ in $V^{mathbb{P}^{op}}$ by mapping an object $X$ of $V$ to the functor $hat{X}: mathbb{P}^{op} to V$ where $hat{X}(0) = X$ and otherwise $hat{X}(n)$ is the initial object $0$. Let $i: V to V^{mathbb{P}^{op}}$ denote this embedding. Then, it is easy to see that the composite

$$V^{mathbb{P}^{op}} times V stackrel{id times i}{to} V^{mathbb{P}^{op}} times V^{mathbb{P}^{op}} stackrel{circ}{to} V^{mathbb{P}^{op}}$$

factors up to isomorphism through the embedding $i: V to V^{mathbb{P}^{op}}$, thus giving a functor

$$V^{mathbb{P}^{op}} times V stackrel{bullet}{to} V$$

where the monoidal category $V^{mathbb{P}^{op}}$ acts on $V$, in such a way that there is a coherent natural isomorphism $(F circ G) bullet X cong F bullet (G bullet X)$. If you work through the details, you find that

$$F bullet X = sum_k F(k) otimes_{S_k} X^{otimes k}$$

This type of structure, where a monoidal category $M$ acts on a category $C$ in this coherent categorified fashion, is called an actegory, and the general nonsense in this case is that a monoid $A$ in $M$ induces a monad on $C$, given objectwise by $X mapsto A bullet X$. In particular, an operad as monoid in $V^{mathbb{P}^{op}}$ induces a monad on $V$, and it's the usual monad on $V$ attached to an operad $A$ with components valued in $V$.

at.algebraic topology - Do "surjective" degree zero maps exist?

It is a theorem of H. Hopf that a map between connected, closed, orientable n-manifolds of degree 0 is homotopic to a map that misses a point, when n > 2. See D. B. A. Epstein, The degree of a map. Proc. London Math. Soc. (3) 16 1966 369--383, for a "modern" discussion including the analogous situation in the non-orientable case. The same result holds for n = 2, but is more difficult and is due to Kneser. See Richard Skora, The degree of a map between surfaces.
Math. Ann. 276 (1987), no. 3, 415--423, for a thorough discussion of the non-orientable case in dimension 2.

at.algebraic topology - Cohomology rings of GL_n(C), SL_n(C)

If $G$ is a connected Lie group (or just a connected loop space with finite homology) then $H^*(G,mathbf{Q})$ is a Hopf algebra. Graded connected Hopf algebras over $mathbf{Q}$ are always tensor products of exterior algebras in odd degrees with polynomial algebras in even degrees. Since polynomial algebras are infinite, they can't occur. The reference to Hopf is probably H. Hopf, Über die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929), 493–524. For the classification of Hopf algebras, see also A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de
groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207.

ag.algebraic geometry - ubiquitous quantum cohomology

"I'd like to know more" is the vaguest possible question :-) The best you can do is to watch the corresponding talk here - it is fantastic ! - and probably come back with more questions, since it is quite dense.

The background for the statement you are asking about is roughly the following (I am no expert and writing from memory - the following may contain nonsense and should not be taken literally):

Consider the moduli stacks $overline{mathcal{M}_{g,n}}$ of stable curves of genus g with n marked points. These are interconnected by a bunch of maps corresponding to glueing together stable curves, forgetting marked points, collapsing components to restore stability etc. (these are all operations with an intuitive geometric meaning and are wonderfully explained in Kock/Vainsencher's "Invitation to Quantum Cohomology").

If we map these into the category of motives, we can for each $n$ form their direct sum over $g$, thus getting a sequence of motives, and then assemble the above maps to form the structure maps of an operad, i.e. mainly "composition maps" $overline{mathcal{M}_{*,n_1}} times ldots times overline{mathcal{M}_{*,n_k}} rightarrow overline{mathcal{M}_{*,n_1 + ldots + n_k}}$.

The word "motives" here means Chow motives and to make sense of this for stacks, one has to extend the definition of Chow groups to suitable stacks and then apply the usual constructions. This can be done in at least two different ways (Vistoli's and Toen's, see the talk for references).

Then the statement in the talk is that the motive of any smooth projective variety ("projective scheme" is a bit too bold because we are using Chow motives) is acted upon by this operad object in motives. To construct the action morphisms one uses cycles (and thus morphisms in motives) $I_{g,n}(V)$ corresponding to certain quantum cohomology classes, which are present in the quantum cohomology of any smooth projective variety...

Of course if you apply now any monoidal functor to all of this (e.g. a cohomology) you get an operad, and an object with an action from it, in the target category...

Edit (Thomas):

Here are Tom Coates' notes.

A try to summarize the seminar talks mentioned below:
Only small quantum cohomology was discussed. The basic idea is to do
the same as with normal motives, i.e. to map the cat. of varieties
into something that looks like a Tannakian category. Whereas usually,
one seeks to classify and identify motives by their corresponding
representations of Gal(Q), here on looks for representations of a
bigger "quantum Galois group of Q" found within known representations
coming from geometry. That "quantum Galois group" was not described,
maybe this relates to it?

Golyshev mentioned as motivating analogy an article by Deligne, where
D. proves the Weil-conjectures for K3-surfaces by embedding motives
from them in a product of motives from abelian varieties, for whom the
Weil-conj.s were known to be true. That embedding of motives was done
by looking how the representations fit together, and by identifying
whose rep.s come from abelian varieties. In Golyshev's analogy, even
dim. quadrics should play the role of K3-surfaces, the orth. group the
role of a construction by Kuga-Satake in Deligne's paper on the repr.s
of ab. var.s

A search method for such representations which would led to "quantum
motives" is made with help of mirror symmetry. That idea seems to go
back to Golyshev.

Coates' method seems to be: The q.-cohomology of Fanos is known, that of toric
varieties is not; On the other hand, the Laurent polynomials of toric
varieties are known, but that of Fanos are not. Mirror symmetry
connects q.-cohomology with such polynomials and Golyshev has a
conjecture conc. the Laurent-poly.s of Fanos. Coates' computations try
to bridge between Fanos and toric varieties
by deformation theory and look if polynomials result which look like
coming from pieces of Fanos, which then could be "quantum motives".

Maniel's idea is to look at the quantum cohomology of Grassmanians
etc., (if I understood him correctly) producing other interesting
representations through a quantum Satake correspondence. Unfortunately
he mentioned that with the quantum Satake so shortly at the beginning
of his talk, that I am unsure if my idea on it's use is correct.

A very fascinating talk was by Gorbounov, he plays around with
Landau-Ginzberg potentials and finds quantum cohomology as special
case of equivariant coh. However that may be, as far as I know, Landau-Ginzberg potentials are only used in Grassmanians etc., q.-coh. is not restricted to them.

One talk I understood nothing from was by Katzarkov, probably similar to his
Oberwolfach talk earlier this year. The connection to quantum motives is unclear to me.

differential equations - Analytical steady-state solution of a complex ODE

I'm a biologist in the process of modeling a fairly simple biological system using a system of ODEs. To verify the simulations, I'm attempting to obtain an analytical steady-state solution that I can check the simulations against. My attempts so far haven't borne fruit, so I thought I'd toss the question out to mathematicians. This is my first post, so apologies if the question isn't right for this site.

The equation is of the form:

$${dS_3over dt} = 2Xv_{max} {S_1 - {S_3^2S_7^4over K_{eq,3}}over K_m+S_1+{S_3^2S_7^4over K_{eq,3}}} + D(S_{3,in} - S_3)$$

$${dS_4over dt} = Xv_{max} {S_1 - {S_4S_7^2over K_{eq,4}}over K_m+S_1+{S_4S_7^2over K_{eq,4}}} + D(S_{4,in} - S_4)$$

$${dS_1over dt} = -Xv_{max} Bigg[{S_1 - {S_3^2S_7^4over K_{eq,3}}over K_m+S_1+{S_3^2S_7^4over K_{eq,3}}} + {S_1 - {S_4S_7^2over K_{eq,4}}over K_m+S_1+{S_4S_7^2over K_{eq,4}}}Bigg] + D(S_{1,in}-S_1)$$

$${dXover dt} = Xv_{max}Y Bigg[4{S_1 - {S_3^2S_7^4over K_{eq,3}}over K_m+S_1+{S_3^2S_7^4over K_{eq,3}}} + 3{S_1 - {S_4S_7^2over K_{eq,4}}over K_m+S_1+{S_4S_7^2over K_{eq,4}}}Bigg] + D(X_{in}-X)$$

$${dS_7over dt} = Xv_{max} Bigg[4{S_1 - {S_3^2S_7^4over K_{eq,3}}over K_m+S_1+{S_3^2S_7^4over K_{eq,3}}} + 2{S_1 - {S_4S_7^2over K_{eq,4}}over K_m+S_1+{S_4S_7^2over K_{eq,4}}}Bigg] + D(S_{7,in}-S_7)$$

Where S₁, S₃, S₄ and S₇ and X are variables

and

K_m, K_eq,3, K_eq,4, v_max, S_1,in, S_3,in, S_4,in, S_7,in, X_in, D and Y are constants.

This system models the change in the substrate S_n or the microbial population X in a perfectly-stirred vessel with microbes acting upon a substrate S₁ to produce S₃, S₄ and S₇ when the kinetics of the chemical reactions are thermodynamically reversible.

S_n,in is the input concentration of S_n. K_m and v_max are constants that describe the "affinity" of the microbe to S₁ and the maximum rate of the reaction respectively and K_eq,n is the thermodynamic equilibrium constant for the reaction S₁ -> A S_n + B S₇. I need to solve this system for S_n where n=1,3,4,7.

Is this even possible, or am I barking up the wrong tree here?

soft question - What are the most overloaded words in mathematics?

-ary

-ary, as in $k$-ary numeral $s$, refers to the number $k$ of values in the domain $K = lbrace 0, 1, ldots, k-1 rbrace$ that affords the basis of numeration.

-ary, as in $k$-ary relation $L$, refers to the number of domains $X_1, ldots, X_k$ for which $L subseteq X_1 times ldots times X_k$.

-ary, as in $k$-ary operation $f$, refers to the number of domains in the domain of the function $f : X_1 times ldots times X_k to Y$, the rubric being, "a $k$-ary operation is a $(k+1)$-ary relation".

Some writers use Greek roots and the Greek suffix "-adic" for the number of domains in a relation, hence medadic, monadic, dyadic, triadic for relations of 0, 1, 2, 3 places, respectively. This usage actually has a degree of historical precedence and it can serve to sidestep conflicts with the domainance of binary numerals in our modern world, but of course the wrinkle but moves to other domains where writers are adicted to other habits.

NB. All puns are intended.

pr.probability - Covariance of points distributed in a n-ball

The covariance is a multiple of the identity by simple symmetry considerations. For the constant, you just need, again by symmetry, and integration in spherical coordinates,

$$mathbb{E} X_1^2 = frac{1}{n} mathbb{E} |X|^2 = frac{c}{n} int_0^R r^{n-1} r^2 dr = frac{c}{n(n+2)}R^{n+2},$$
where $R$ is the radius of your ball and $c$ is a constant depending on $n$ and $R$. To identify $c$,
$$1 = c int_0^R r^{n-1} dr = frac{c}{n} R^n,$$
so $c = n/R^n$ and your covariance is $frac{1}{n+2} R^2$ times the identity matrix. Hopefully I've included enough detail that if I've made an algebra mistake it will be easy for someone else to correct it, but I think I recognize that as the right answer.

soft question - How have mathematicians been raised?

Here are interesting reviews of russian math circles. Friends who experienced such math circles (in the USSR and copies in other countries) told that it works great, even if one later decides for not becoming a mathematician.

Gromov expressed in his interview here some very interesting thoughts: "Now there are no more Abels... It means that they have been destroyed. The education destroys these potential geniuses – we do not have them! ... There are some experiments on animals which indicate that the way you teach an animal is not the way you think it happens. The
learning mechanism of the brain is very different from how we think it works: like in physics, there are hidden mechanisms. We superimpose our view from everyday experience, which may be completely distorted. Because of that, we can distort the potentially exceptional abilities of some children. ... There are very interesting experiments performed with Chimpanzee and Bonobo apes and under which conditions they learn, or even how you teach a parrot to talk. How do you do that? The major factor is that it should not see the teacher. You put a mirror between you and the parrot and then you speak behind the mirror. The parrot then sees a bird – it talks to a bird. But if it sees you, it will learn very badly. That is not an obvious thing. The very presence of a teacher, an authority, moves students in a particular direction and not at all the direction the teacher wants them to move. With all this accumulated evidence, you cannot make any simple decision."

Edit: Alexandre Borovik links in his blog to this very fascianating essay by Gromov: "We introduce a concept of an ergosystem which functions by building its ”internal structure“ out of the ”raw structures“ in the incoming flows of signals. ... We shall argue in this section that the essential mental processes in humans andhigher animals can be represented by an abstract universal scheme and can bestudied on the basis of few (relatively) simple principles."

In general, I guess one should have the time frame of brain development in mind: One 'developmental window' is ca. from birth to age 4, the other around puberty. In both, many new nerves/nerve-connections build up, but untrained ones are then deleted. Reports of recent studies suggest that pre-birth risks fro the brain like stress, lead exposure, malnutrition are important too and widely underestimated.

Comes the dawn of 'Algernon mathematicians'
?

Edit: Here is the link to an article about a project to identify "good teaching": "The most stunning finding to come out of education research in the past decade: more than any other variable in education—more than schools or curriculum—teachers matter. But we have never identified excellent teachers in any reliable, objective way."

fourier analysis - level sets of multivariate polynomials

Let $p:mathbb R^n rightarrow mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^nsubsetmathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$:

$$int_Q p(x) dx = 0.$$

For $alpha>0$ consider the sublevel sets of $P$,

$$E_alpha= {xin Q: |p(x)|leq alpha}$$

There are several known estimates for the Lebesgue measure of this set which in some sense or another are uniform over some classes of polynomials. For example, we have that

$$|E_alpha| lesssim min(pd,n) frac{ alpha^{1/d} }{ |p|_{ L^p(Q) }^{1/d} }$$

This particular estimate is due to Carbery and Wright and can be found here.

I'm interested in studying the (induced Lebesgue) measure of the boundary of this set

$$|partial E_alpha|=|{xin Q: |P(x)|=alpha}|$$

Consider first the easy case of dimension $n=1$. Then the set $E_alpha$ is a finite union of closed intervals and the question is trivial. It is obvious that in this case there are at most $O(d)$ intervals so the $0$-dimensional measure of the boundary is $O(d)$.

Now in many variables things will be much more complicated. For example can we say that the set $E_alpha$ has $O(d)$ connected components? Is there an estimate for the measure of the boundary $partial E_alpha$ in terms of $alpha$, $d$ and $n$, assuming (say) that $|p| _ {L^1(Q)}=1$ ?

This question comes up naturally if one tries to study an oscillatory integral with phase $p$ and apply integration by parts (i.e Gauss theorem) imitating the one dimensional method of proving the van der Corput lemma (for example).

ag.algebraic geometry - Matrix factorization categories for ADE singularities

Kevin: C[x,y] is naturally Z-graded by deg x = 1, deg y = 1. this induces a Z-grading on the ring of invariants A = C[x,y]^G. If G is not cyclic of odd order, then A is supported in even degrees, i.e. A_n = 0 for n odd. this is the natural grading on A. One often changes the grading by A_n = A_{n/2}; this is called the reduced grading.

There are also fractional gradings: Write C[x,y] = Sym V, where V is a two dim'l vector space. Let R = C[x^2,xy,y^2] = C[u,v,w]/uv-w^2. If we set deg (u,v,w) = (1,1,1), then uv-w^2 is homogeneous. then x and y have degree "1/2".

Incidentally, this is what all the "Spin 1/2" business is about. V is called D_{1/2}. V^{otimes 2j} is called D^j. The Clebsch-Gordan formula tells one how to decompose tensor powers of V. It says D^j otimes D^k = D^{abs j -k} oplus D^{abs {j - k}+1} oplus ... oplus D^{j+k}. see Varadarajan Supersymmetry chapt 1.

ag.algebraic geometry - Moduli space of K3 surfaces

It is known that there exists a fine moduli space for marked (nonalgebraic) K3 surfaces over $mathbb{C}$. See for example the book by Barth, Hulek, Peters and Van de Ven, section VIII.12. Of course the marking here is necessary, otherwise the presence of automorphisms can be used to construct isotrivial non trivial families of K3.

Assume that we want to construct an algebraic analogue of this. Of course some structure has to be added; I am thinking of something like torsion points for elliptic curves.

Is there a way to add some structure and actually build a fine moduli space of (K3 + structure) which is defined over $mathbb{Z}$?

Of course this will only parametrize algebraic K3 surfaces, so it will need to have infinitely many components. A second question, if we do not want to add structure, is

Is there a fine moduli stack of algebraic K3 surfaces which is algebraic (either in the DM or Artin sense)?

It is not difficult to produce K3 surfaces with a denumerable infinity of automorphisms, so I'm not really expecting the answer to be yes, but who knows.

Edit: since there seems to be some confusion in the answers, the point is that I'm asking for a FINE moduli space. I'm aware that one can consider moduli spaces of polarized K3, and this is why I wrote "Of course this will only parametrize algebraic K3 surfaces, so it will need to have infinitely many components."

ag.algebraic geometry - What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve?

Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X times ... times X$ by the symmetric group $S_n$). There is a well known, cool formula computing the Euler characteristic of all these n-symmetrical products:

$$sum_{d geq 0} chi left(X^{[d]} right)q^d = (1-q)^{- chi(X)}$$

It is known that $S^nX cong X^{[n]}$, the Hilbert scheme of 0-subschemes of length n over $X$. Hence, the previous formula also computes the Euler characteristic of these spaces.

What about for singular surfaces? More precisely, if $X$ is a singular complex algebraic curve, do you know how to compute the Euler characteristic of its n-symmetrical powers $S^nX$? More importantly: what is the Euler characteristic of $X^{[n]}$, the Hilbert scheme of 0-schemes of length n over $X$?

I guess it is too much to hope for a formula as neat as the one given for the smooth case. Examples, formulas for a few cases or general behaviour (e.g. if for large n, $chileft(X^{[n]}right) = 0)$ are all very welcome!

ag.algebraic geometry - How do you see the genus of a curve, just looking at its function field?

Let $k$ be the ground field.

The Kahler differentials $Omega_{K/k}$ are the $K$ vector space generated by formal symbols $dg$ subject to $d(f+g)=df+dg$, $d(fg) = f dg + g df$ and $da=0$ for $a in k$. This is a one dimensional $K$ vector space.

Let $omega$ be a differential. For any valuation $v$ on $K$, let $t$ be such that $v(t)=1$. We say that $omega$ is regular at v if $v(omega/dt) geq 0$. We say that $omega$ is regular if it is regular at every valuation of $K/k$. Then the space of regular differentials is a $k$ vector space of dimension $g$.

FURTHER THOUGHTS

Both Ben's answer and mine used the set of valuations of $K/k$. This essentially means that we used the ground field $k$. A valuation of $K/k$ is defined as a valuation of $K$ which is trivial on $k$; conversely, the ground field can be recovered from the valuations that respect it by the formula $k = bigcap_{v} v^{-1}(mathbb{R}_{geq 0})$.

Here is a cautionary example to show that there can not be any solution which only uses properties of the field $K$, without reference to $k$. Let $C$ and $D$ be two irreducible curves of different genuses. Let $K$ be the field of meromorphic functions on $C times D$, let $k$ and $ell$ be the fields of functions on $C$ and $D$. Then $K$ is a transcendence degree 1 extension of both $k$ and $ell$, but has different genuses when considered in these two ways.

ag.algebraic geometry - Is every subgroup of an algebraic group a stabilizer for some action?

Suppose G is an algebraic group (over a field, say; maybe even over ℂ) and H⊆G is a closed subgroup. Does there necessarily exist an action of G on a scheme X and a point x∈X such that H=Stab(x)?

Before you jump out of your seat and say, "take X=G/H," let me point out that the question is basically equivalent^† to "Is G/H a scheme?" If G/H is a scheme, you can take X=G/H. On the other hand, if you have X and x∈X, then the orbit of x (which is G/H) is open in its closure, so it inherits a scheme structure (it's an open subscheme of a closed subscheme of X).

^†I say "basically equivalent" because in my argument, I assumed that the action of G on X is quasi-compact and quasi-separated so that the closure of the orbit (i.e. the scheme-theoretic closed image of G×{x}→X) makes sense. I'm also using Chevalley's theorem to say the the image is open in its closure, which requires that the action is locally finitely presented. I suppose it's possible that there's a bizarre example where this fails.

nt.number theory - How to put "x^a => 1/(a+1) x^(a+1)" and "x^-1 => log(x)" together

First of all, I'll repeat what I said in the comments: writing $frac{x^h - 1}{h} = frac{e^{h log x} - 1}{h}$ makes it fairly clear that as $h to 0$, this expression tends to $log x$, and setting $h = a + 1$ this is precisely the desired result.

I should mention that this result is implicit in a certain fact well-known to people who do competition math, which is as follows. Given non-negative real numbers $x_1, ... x_n$, let $A_p(x_1, ... x_n) = sqrt[p]{ frac{x_1^p + ... + x_n^p}{n} }$ denote the $p$-power mean for $p neq 0$. For $p = 0$, define $A_0(x_1, ... x_n) = sqrt[n]{x_1 ... x_n}$ (the geometric mean, and also the limit as $p to 0$ of the above).

Theorem (Power Mean Inequality): If $p le q$, then $A_p le A_q$.

If you like fancy keywords, then I will bring to your attention that as $p to infty$ the $p$-power mean approaches $text{max}(x_1, ... x_n)$, which one can think of as the "low-temperature limit" of ordinary addition becoming tropical addition. Then $p to 0$ can be thought of as the "high-temperature limit," in which ordinary addition becomes multiplication instead. Somebody who knows more statistical mechanics than I do (that is, any) can probably tell you the physical significance of this.

nt.number theory - A local-to-global principle for being a rational surface

It seems to me that there are irrational surfaces over $mathbb Q$ that are $mathbb Q_v$-rational for all $v$. (I couldn't find them in the literature, but didn't look very hard. Almost certainly they are to be found there, in papers by either Iskovskikh or Colliot-Thelene.)

Take the affine surface $S$ given by $y^2+byz+cz^2=f(x)$, where $f$ is an irreducible cubic and $b^2-4c$ equals the discriminant $D(f)$ of $f$, up to a square in $mathbb Q^*$, and $D(f)$ is not a square. According to Beauville-Colliot--Thelene--Sansuc--Swinnerton-Dyer $S$ is not $mathbb Q$-rational, but is stably rational. (Irrationality is Iskovskikh I think, in fact.) Via projection to the $x$-line a projective model $V$ of $S$ is a conic bundle over $mathbb P^1$ with $4$ singular fibers (one is at infinity). There is an embedding of $V$ into a weighted projective space $mathbb P(2,2,1,1)$; the defining equation is $Y^2+bYZ+cZ^2=F(X,T)T$, where $F$ is the homogeneous version of $f$. By construction the Galois action on the $8$ lines that comprise the singular fibers is via the symmetric group $S_3$: the two lines in the fiber at infinity are conjugated, and the other six are permuted transitively.

Claim: Assume that $D(f)$ is square-free and prime to $6$. Then $S$ is $mathbb Q_v$-rational for all $v$.

Proof: Suppose that the decomposition group $G_v$ at $v$ is cyclic. Whatever its order ($1,2$ or $3$) there are at least $2$ disjoint lines among the $8$ that are $G_v$-conjugate, so they can be blown down to give a conic bundle over $mathbb P^1$ with at most $2$ singular fibres and a $mathbb Q_v$-point; it is well known that such a surface is $mathbb Q_v$-rational.

Now suppose that $G_v= S_3$. Then $v$ is non-archimedean and $V$ has bad reduction there. In fact, exactly two of the singular fibers are equal modulo $v$; it follows that $G_v=S_3$ is impossible, and we are done.

E.g., $f=x^3+x+1$, of discriminant $-31$, $c=8$, $b=1$.

(This doesn't use stable rationality, but rather the fact that these surfaces, although irrational, are very close to being rational, in the sense that the action of $Gal_{mathbb Q}$ on the lines is as small as possible subject to the surface being irrational, and the action of the decomposition groups is even smaller.)